Free Fields Associated with the Relativistic Operator
−(m − √
m 2 − ∆)
Narn-Rueih Shieh
†Abstract
The purpose of this article is to consider the construction of free fields associ- ated with the relativistic operator −(m −√
m2− ∆). The study is based on the viewpoint of pseudo-differential operators, and we present both the Gaussian case and the non-Gaussian infinitely divisible case. We prove that, in the Gaussian case our constructed field is singular with respect to the Gaussian free field based on the non-relativistic m2− ∆ [15].
1 Introduction
The purpose of this article is to consider the construction of free fields associated with the relativistic operator
−∆(r)m := −(m −√
m2− ∆),
where m > 0 is the normalized mass of a relativistic particle. On the one hand, the operator −∆(r)m is the so-called free relativistic Hamiltonian, and is key to the physical theory of stability of matter, pioneered by Lieb [7] in 1970’s. A seminal paper by Car- mona et al. [3] investigated the mathematical theory of this operator, and in particular
†Mathematics Department, Honorary Faculty, National Taiwan University, Taipei 10617, Taiwan.
E-mail: [email protected] URL: http://www.math.ntu.edu.tw/~shiehnr/
its close relation to L´evy processes; see also [12, 1] for the subsequent studies, including a fractional version. A very recent article [4] contains the sharp estimate of the associ- ated heat kernel and also useful references. The operator has interesting multi-scaling property; see [17, 9]. On the other hand, Gaussian free fields (GFFs, for brevity) have been understood as the building block of quantum field theory, and is also key to the recent advent in quantum gravity [5] and the Schramm-Loewner evolution [14], see the inspiring survey of Sheffield [15]. The GFF in [15] is mainly massless, and the mas- sive case mentioned in Section 3.3 there is based on m2 − ∆ (for which we may say, non-relativistic).
We aim to construct free fields associated with −∆(r)m from the viewpoint of the theory of pseudo-differential operators; see, for example, the book of Wong [21] for a basic treatment of the theory. We present both the Gaussian case and the non-Gaussian infinitely divisible case. We prove that, in the Gaussian case our field constructed is singular with respect to the GFF based on non-relativistic m2 − ∆.
We present our results in Section 2, and all the proofs are given in Section 3.
Acknowledgement: This work is initialized while the author visited Department of Mathematics, The Chinese University of Hong Kong, in Spring 2012. The hospitality of the host is appreciated. I am indebted to K.S. Lau for helpful discussion on the analytic theory of Green’s functions, and to M.W. Wong for helpful discussion ( via emails) on the symmetry (the self-adjointness) of the operator −∆(r)m , related to [20]. I also thank to the referee for her/his very careful reading and constructive comments, which make the present version much more stylish and edged.
2 Main results
Since we treat −∆(r)m as a pseudo-differential operator, as that in [3] and [20]; we work on the whole Euclidean space Rn, n ≥ 2, and is in the framework of tempered dis- tributions. One may refer to, say, the book [21] for basic notions and properties of
pseudo-differential operators and tempered distributions. The so-called restriction prob- lem of free fields will be addressed elsewhere. Thus the space of test functions is the Schwartz space S(Rn), consisting of all rapidly decreasing functions, equipped with the Schwartz topology for such functions (somewhat different for those C∞of compact sup- ports); see [21]. Meanwhile, for an f ∈ L2(Rn, Leb), we use ˆf or F f to denote the Fourier (-Plancherel) transform of f . For an f ∈ S(Rn), define ∆(r)m f via the Fourier transform
\∆(r)m f (ξ) = θ(ξ) ˆf (ξ), ξ ∈ Rn, with
θ(ξ) := (m2+ |ξ|2)1/2− m > 0, ∀ξ 6= 0.
The following proposition is the starting point of our study
Proposition 1. For f, g ∈ S(Rn), the operator ∆(r)m defines an inner product on Rn by (f, g)(r)m := (f, −∆(r)m g) = (−∆(r)mf, g),
where (·, ·) denotes the usual inner product on L2(Rn, Leb).
We observe that, since θ(ξ) ' |ξ| as the latter is large, the Hilbert-space closure of S(Rn) under the inner product (f, g)(r)m is the Sobolev-Bessel space (see, for example, [21], or the more advanced [19]).
H1,2(Rn) = {f ∈ L2(Rn, Leb) : F−1(1 + |ξ|2)1/2(F f )(ξ) ∈ L2(Rn, Leb)}.
The following notion is essential to our study; it is natural to call the positive quantity (f, f )(r)m for each nonzero f ∈ S(Rn), or for more general nonzero f ∈ H1,2(Rn), to be the relativistic energy of f , and denote it by Em(r)(f ). The notion corresponds to the (traditional) Dirichlet energy, denoted by E∇(f ), based on the Dirichlet product (f, f )∇; see [15, Section 2.1]. Now, we present the following definition:
Definition. Given an underlying complete probability space (Ω, P ), the free field associated with −∆(r)m , denoted by Xm(r), is the unique linear random functional (Xm(r), f ),
indexed by f ∈ S(Rn) (the extension to more generally f will be discussed below), taking values in L2(Ω, dP ) (the Hilbert space of square-integrable random variables defined on Ω), and continuous in the sense that:
fn → 0 in S(Rn) ⇒ (Xm(r), fn) → 0 in L2(dP ).
The random variables (Xm(r), f ) are required to be centered (i.e. mean zero) and with the covariance structure
E[(Xm(r), f )(Xm(r), g)] = (f, g)(r)m ;
in the above, the notation E[ · ] denotes the expectation, i.e. the mean (the integral), of a random variable with respect to the underlying probability measure P . Moreover, we assume that each (Xm(r), f ) is an infinitely divisible random variable, either Gaussian or non-Gaussian.
We should remark that the above definition is consistent with the existing litera- ture, as follows.
Firstly, by Proposition 1, (f, g)(r)m is a positive-definite bilinear form, it is qualified to be a covariance structure, and therefore the above definition is indeed a kind of generalized random field in the classic paper by Yaglom [22] on the correlation theory (spectral analysis) of L2 random fields; this paper is mainly in the context of generalized random fields, i.e. linear random functionals, except he used the C∞ functions with compact supports as test functions. Secondly, in the theory of correlation theory, it is mostly in the framework of the second moment, i.e. for random variables with the L2(dP ) norm; while this article, as we have imposed in the definition, we assume further that each (Xm(r), f ) is an infinitely divisible (ID, for short) random variable. Then, due to the linearity assumption, each finite linear combination a1(Xm(r), f1) + · · · + ak(Xm(r), fk) is also an ID random variable. Therefore, the family of random variables (Xm(r), f ), indexed by f , is then a centered ID system with second moments; see Rajput and Rosinski [10] for the intensive study of the structure of such ID systems. Thirdly, and most prominently, in the Gaussian case our proposed definition in above is seen as a one mentioned in [6, Example 1.16, p. 7].
An ID random variable is characterized by its L´evy triplet (see, for example, Sato [13]); the drift (which is zero since our field is centered), the Gaussian part, and the jump-part (which is determined by its L´evy measure ν(dx)). We treat the Gaussian and the non-Gaussian separately. Write, under the ID assumption, the characteristic function (chf) as
ψ(s, f ) := E[eis(Xm(r),f )] := e−k(s,f ), s ∈ R,
where k(s, f ) is the L´evy-Khintchine representation; we treat separately as
∗ Gaussian: k(s, f ) = c(f )s2, where c(f ) is a positive constant to be determined.
∗∗ Non-Gaussian:
k(s, f ) = Z
R
[eisx− 1]ν(dx, f ),
where the L´evy measure ν is to be determined; we remark that, since we have assumed that (Xm(r), f ) has second moment, it is legitimate and convenient to use the integral representation for k(s, f ) in this form. We have
Proposition 2. Let Xm(r) be an ID free field associated with −∆(r)m, as defined above, for which the L´evy-Khintchine representation of (Xm(r), f ) is e−k(s,f ). Then,
∗ Gaussian: k(s, f ) = c(f )s2, with c(f ) = Em(r)2(f ).
∗∗ Non-Gaussian: k(s, f ) =R
R[eisx− 1]ν(dx, f ), with L´evy measure ν(dx, f ) satisfy- ing the scaling, for each nonzero a,
ν(d(x/a), f ) = ν(dx, af ).
Remark: In a subsequent paper [18], we choose ν(dx, f ) to be with the density q(x, f )dx, where
q(x, f ) = 1 4π
1 q
Em(r)(f ) Z ∞
0
h 1
√u· e−
x2 4uE(r)
m (f )
ihe−m2u u32
i du,
for which the required scaling in Proposition 2 is satisfied.
In theory of free fields, it is of essential importance to extend (Xm(r), f ) defined for more general f other than f ∈ S(Rn); see [15] for the detailed discussion of the (massless)
GFFs based on the Dirichlet product, (f, f )∇(the massive case also included). We treat the Gaussian and the non-Gaussian separately. We assume Proposition 2 to be valid.
In the Gaussian case,
E[eis(Xm(r),f )] := eE
(r) m (f )
2 s2.
Then, (Xm(r), f ) is consequently defined for f ∈ H1,2(Rn). The latter space is the Hilbert closure of S(Rn) under the (f, g)(r)m, as we have observed in above; therefore, we have the Gaussian-Hilbert space [6, Definition 1.19, §3, Chapter 1] based on the relativistic energy Em(r)(f ). One should compare it with the treatment in [15] based on the Dirichlet energy E∇(f ).
The main contribution of this article is to present the following “Green’s function representation” for the free field defined above, including both the Gaussian and the non-Gaussian case. We should remark that, the following theorem appears to be new, to our knowledge, even in the Gaussian case.
Theorem 1. When n ≥ 3, there is a unique nonnegative function, for which we call the relativistic Green’s function G(r)m(x, y) associated with ∆(r)m, such that G(r)m (x, y) = G(r)m (y, x) and that, for f, g ∈ S(Rn),
(f, g)(r)m = Z Z
f (x)G(r)m (x, y)g(y)dxdy = Z Z
f (y)G(r)m (y, x)g(x)dydx. (2.1) For n = 2, it has to be confined to x, y ∈ D(0, R), an open disc with center at origin and with radius R ( R can be arbitrarily large, yet needs to be finite); equivalently, it has to assume that f, g are supported on D(0, R).
Symbolically, we may say that G(r)m (x, y) solves, in the sense of tempered distribu- tions,
∆(r)m G(r)m(x, ·) = −δx(·);
we confer this notion to the Green’s function for the classical Laplacian in, say, [8, p.
148].
Here, the dimensionality n is crucial: for n ≥ 3, G(r)m (x, y) can be, as we will do in the following context, defined for all x, y ∈ Rn; however, for n = 2, we need f, g to be
supported on inD(0, R), an open disc with center at origin and with radius R ( R can be arbitrarily large, yet needs to be finite). The proof of the above Theorem 1 (given in Section 3) is mainly based on the probabilistic viewpoint, in which the dimension n = 2 has to be separated. The author is not able, it would be very interesting, to give an wholly analytic proof.
Remark: It is here an occasion to mention a terminology in [17]. In that article, the notation Gt(x, y) is also referred as a Green’s function, although it is better known as the heat kernel associate with ∆(r)m .
We extend, as a consequence, (Xm(r), f ) to f ∈ B+(Rn), the space of bounded nonneg- ative functions, and the equation (2.1) in Theorem 1 thus holds for all f, g ∈ B+(Rn);
in the n = 2, we need to confine f, g being supported on D(0, R).
We remark that the Green’s function representation is crucial to employ GFFs to both the quantum gravity [5] and the Schramm-Loewner evolution [14]. We mention the Gaussian-Hilbert setting in the Gaussian case, mainly for the following
Theorem 2. The Gaussian-Hilbert space based on the relativistic energy Em(r)(f ) and that one based on the Dirichlet energy E∇(f ) are mutually singular.
We will prove this “natural” result by the Gaussian dichotomy, see (for example) [2, Chapter 2], and by an early result of Shepp [16] (there are a lot of refinements on the Gaussian dichotomy in vast literature, admittedly).
Remark: In general, the existence of “Green’s function” in other related models is a core issue; see a very recent article on Kahane’s theory of multiplicative chaos [11].
3 Proofs
Proof of Proposition 1. Define, for each t > 0, a kernel Kt(x), x ∈ Rn, via its Fourier transform by
Kˆt(ξ) = 1
√2π
n
e−tθ(ξ).
The kernel Kt(x) is symmetric Kt(x) = Kt(−x), and is radial Kt(x) = Kt(||x||). We then define a semigroup of operators on L2(Rn) by
Tt(f ) = (Kt∗ f )(x) = Z
Kt(x − y)f (y)dy, T0f = f.
We claim that {Tt} is a contractive semigroup: kTtf kL2(Rn) ≤ kf kL2(Rn); by the Fourier transform for convolutions and the Plancherel identity, we have,
Z
x
(Ttf )2(x) dx = Z
ξ
| ˆKt(ξ) ˆf (ξ)|2 dξ
= Z
ξ
e−tθ(ξ)f (ξ)|ˆ 2
≤ Z
ξ
| ˆf (ξ)|2 dξ = Z
x
f2(x) dx.
Now, we prove that, for each φ ∈ S(Rn), k(Tt− I)φ
t + ∆(r)mφkL2(Rn) → 0, as t ↓ 0,
which means that, c.f. [20, Theorem 4.1], the infinitesimal generator of the L2 semigroup {Tt} is −∆(r)m, at least acting on S(Rn). To prove the above display, write Dt(φ) =
(Tt−I)φ
t + ∆(r)m φ, and observe that Z
(Dtφ)2(x)dx = Z
| dDtφ|2(ξ)dξ
=
Z he−tθ(ξ)− 1
t + θ(ξ)i2
φ(ξ)ξ.ˆ
For each ξ, the term in the above bracket tends to 0 as t ↓ 0; hence the whole integral in the above must also tend to 0, since ˆφ(ξ) is rapidly decreasing in ξ.
For each t > 0, we have the symmetric (the self-adjoint) property of Tt on S(Rn):
(f, Ttg) = (Ttf, g), which follows from the symmetry of the kernel Kt(x). Thus, so is for the operator −∆(r)m , i.e. (f, −∆(r)m g) = (−∆(r)m f, g). Moreover, for each f ∈ S(Rn),
Z
(−∆(r)mf )(x) · f (x)dx =
Z \∆(r)mf (ξ) · ˆf (ξ)dξ = Z
θ(ξ)| ˆf (ξ)|2dξ ≥ 0,
since θ(ξ) > 0, ∀ξ 6= 0; the above equals to 0 only when f = 0. Therefore, the (f, g)(r)m
indeed defines an inner product on S(Rn).
Proof of Proposition 2. Observe that the chf ψ(s, f ) := E[eis(Xm(r),f )] := e−k(s,f ) must satisfy
ψ0(s, f )|s=0= iE[(Xm(r), f )], ψ00(s, f )|s=0 = −E[(Xm(r), f )2].
Thus, for the Gaussian case, in which k(s, f ) = c(f )s2, it follows that c(f ) = E[(Xm(r), f )2]/2 = Em(r)(f )/2.
As for the non-Gaussian case, the k(s, f ) = R
R[eisx − 1]ν(dx, f ), with L´evy measure ν(dx, f ), gives, c.f. [13, p.163],
Z
R
xν(dx, f ) = 0, Z
R
x2ν(dx, f ) = E[(Xm(r), f )2].
The scaling property of the L´evy measure then follow from the linear requirement of Xm(r); namely a(Xm(r), f ) = (Xm(r), af ).
Proof of Theorem 1. The following construction is already noted in [3]; see also [12, 4]. Let B(t) be the n-dimensional standard Brownian motion, and let the so-called relativistic 12-subordinator be defined as a L´evy process Tt with increasing sample paths and with the Laplace function determined by
Eh e−uTti
= e−t(
√
m2+u−m), u > 0.
Assume that the process Tt and the Brownian motion Bt are totally independent, then the subordinated process Xt= BTt is a L´evy process in Rn, for which the characteristic function is given by,
Eh
ei<ξ,Xt>i
= e−t(
√
m2+|ξ|2−m)
, ξ ∈ Rn.
The transition density function p(t, x, y) of the process Xt is then existent, jointly con- tinuous in (t, x, y), and p(t, x, y) = p(t, 0, y − x). The estimate of p(t, x, y) given in [3], see also the latest sharp one in [4] (with α = 1 there), tells that, for n ≥ 3, the
“probabilistic Green’s function”
G(x, y) :=
Z ∞ 0
p(t, x, y)dt,
is defined, finite-valued and symmetric in x, y ∈ Rn.
We prove that this “probabilistic Green’s function” G(x, y) satisfies (2.1). Firstly, the defining property of G(x, y) asserts that G(x, y) = G(y, x) = G(0, x − y) = G(0, y − x), since p(t, x, y) has such spatial homogeneity for each t. Next, comparing the characteris- tic function of Xt given in above and the Fourier transform of the kernel Ktin the proof of Proposition 1, we see that Kt(x) = cn· p(t, 0, x), where cn is an absolute constant depending only on the dimension n. Therefore, the symmetric (the self-adjoint) property of Tt on S(Rn): (f, Ttg) = (Ttf, g), which is mentioned in the proof of Proposition 1, can be re-written as, for t > 0 and for f, g ∈ S(Rn),
Z Z
f (x)p(t, x, y)g(y)dxdy = Z Z
f (y)p(t, y, x)g(x)dydx.
Integrating the above over t ∈ (0, ∞), we have Z Z
f (x)G(x, y)g(y)dxdy = Z Z
f (y)G(y, x)g(x)dydx;
we remark that it is legitimate to change the order of the integration, since G(x, y) has been defined as a nonnegative finite-valued symmetric function in x, y ∈ Rn. To see the above is equal to the (f, g)(r)m, and thus have the equation (2.1), with G(r)m (x, y) being this
“probabilistic Green’s function” G(x, y), we use the same argument as that in [8, p. 166]
for the Laplacian ∆ and the Gaussian heat kernel; it is also parallel to some argument in the proof of [20, Theorem 4.1]. Indeed, −∆(r)m is shown to be the infinitesimal generator of the semigroup {Tt} in the proof of Proposition 1, and thus Ttg = e−t∆(r)mg, at least for g ∈ S(Rn). Thus, we have
− ∆(r)mTtg = d
dtTtg; (3.1)
confer to [8, p. 166]. We recall, from the proof of Proposition 1 and the relation of Kt(x)& p(t, x, y) in above, that
Ttg(x) = (Kt∗ g)(x) = Z
Kt(x − y)g(y)dy = Z
p(t, x, y)g(y)dy, T0g = g.
We then have
(f, e−t∆(r)m g)Leb(dx)= (f, Ttg)Leb(dx)= Z Z
f (x)p(t, x, y)g(y)dxdy.
Integrating the above over t ∈ (0, ∞), we have Z Z
f (x)G(x, y)g(y)dxdy = (f, (∆(r)m )−1g)Leb(dx)= (−∆(r)m f, g)Leb(dx),
which is the left-handed side of (2.1). In the second term of the above, the inverse of the operator ∆(r)m is implied by (3.1). This completes the proof of the theorem for n ≥ 3.
For the n = 2, as in [12, 4], we need to consider, not the whole process Xt but the
“subprocess” killed upon exiting D(0, R), i.e. XtR := Xt, t < τR = inf{t ≥ 0; Xt 6∈
D(0, R)}, = ∂, t ≥ τR, and consider the residual pR(t, x, y) which is p(t, x, y)−rR(t, x, y) (rR is the hitting probability density, see, for example, [12, p. 9]); the corresponding Green’s function is then
G(x, y) := GR(x, y) :=
Z ∞ 0
pR(t, x, y)dt.
The killing is mandate for n = 2, due to the standard fact that the planar Brownian motion is recurrent; while it is transient if the dimension is higher. Therefore, the (2.1) in the dimension n = 2 must need f, g to be supported on D(0, R).
Remark: The proof in above, as we may have seen, blends the analysis and proba- bility, and it hinges on that the probabilistic method in [12, 4] gives the proper estimate to assert the integrability of Kt(x) = cn· p(t, 0, x) over t ∈ (0, ∞).
Proof of Theorem 2. From the Gaussian dichotomy, the two (Xm(r), f ) and (X, f ) must be either mutually singular or mutually equivalent. Notice that the Gaussian- Hilbert space (Xm(r), f ) is for f ∈ H1,2(Rn) while the Gaussian-Hilbert space (X, f ) is for f ∈ H2,2(Rn) (see the Section 2 for the first, and the [15] for the latter), and that H2,2(Rn) H1,2(Rn). For any
f ∈ H1,2(Rn) \ H2,2(Rn),
which means that the relativistic energy Em(r)(f ) is finite and the (traditional) Dirichlet energy E∇(f ) is infinite, it must have, in the sense of probability distributions, (X, f ) ∼
δ0 (degenerate) while (Xm(r), f ) ∼ N (0, Em(r)(f )) (properly Gaussian distributed). This show that the two Gaussian measures cannot be mutually equivalent, and hence must be mutually singular.
We can also prove the mutual singularity of (Xm(r), f ) and (X, f ), for f indexed by the Schwartz space S(Rn), as follows. We apply Shepp’s [16] result in the following form.
Let {Hl(r), l = 0, 1, 2, ...} be the Hermite polynomials, i.e.
Hl(x) = (−1)lex22 dl
dxle−x22 , for l ∈ {0, 1, 2, ...}, x ∈ R.
Thus, the Hermite functions hl(x) := 1
√4
2π
√1
n!e−x24 Hl(x), l = 0, 1, 2, ..., are in S(R1), and form an orthonormal basis for L2(R1, dx). Therefore,
h¯k(x) := hk1(x1) · · · hkn(xn), ¯k = (k1, · · · , kn), x = ((x1, · · · , xn), form an approximating family in S(Rn). We write, for l = 0, 1, 2, ...
πl:= {h¯k(x), ¯k = (k1, · · · , kn), ki ≤ l, ∀i}.
Let ρi, i = 0, 1, denote the covariance matrix associated, respectively, with (X, h¯k) and (Xm(r), h¯k), and ρ be the average matrix ρ = ρ0+ρ2 1. The Hellinger functional up to the order l, H(πl), see [16, p. 169 (1.8)] (remark that we are now in the centered, that is mean zero, and thus there is no exponential part in his (1.8)), is then
H(πl) = (det ρ0(πl))1/4(det ρ1(πl))1/4 (det ρ(πl))1/2 .
The reciprocal of its 4th power is the product of l strictly positive numbers,Ql j
λj+1 2 ·1+λ
−1 j
2
[16, p.169 (1.10), p.170 (1.16)]. Each λj itself is split as λj = δj,0δj,1−1, where δj,i (j = 0, 1, 2, . . .) is the eigenvalues corresponding to ρi. The sequence of eigenvalues of the covariance matrices ρi defined above is unbounded, and thus the l products in the above tend to ∞ as l ↑ ∞. Consequently, the Hellinger functional H of the two Gaussian free fields is zero, and thus they mutually singular.
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